Prescribed modulus chirp-like waveforms with multiple frequency notches

ABSTRACT

An iterative method for modifying an initial time signal to form a created signal having a prescribed envelope, and frequency notches at prescribed frequency values, wherein the created signal closely resembles the initial time signal, the envelope of the created time signal is the prescribed envelope, and the Fourier magnitude of the created time signal at the prescribed frequency values is nearly zero. The created time signal may be a real-valued signal as well as a complex-valued time signal which closely resembles an arbitrary initial time signal, including initial time signals which are standard transmit signals for radar systems, and which have Fourier transform magnitudes with notches and stop-bands at prescribed frequency values. These notches and stop bands are created by enforcing nulls of prescribed order at the prescribed frequency values within the modified time signal.

STATEMENT FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

The present invention is based upon work supported and/or sponsored bythe Information Innovation Office, Defense Advanced Research ProjectAgency (DARPA), Arlington, Va., under SBIR Phase II contract No. D11PC20007.

FIELD OF THE INVENTION

The present invention relates to radar systems, and to methods andapparatus for transmitting signals in radar systems.

BACKGROUND OF THE INVENTION

In radar systems, an electromagnetic signal typically having a pulsedwaveform is transmitted from a radar antenna, reflected from a target ortargets, and received at a radar receiver. The range resolution of theradar system improves with increasing bandwidth of the electromagneticsignal transmitted by the radar. Because typical radar systems mayoperate at frequencies which include reserved or restricted frequencybands, high-bandwidth electromagnetic signal transmittal may interferewith communications, navigation, or other uses of these frequency bands.In particular, if other systems operate in relatively narrow frequencybands, the wide-band spectrum of the transmitted electromagnetic signalwaveform should have notches or stop-bands at the respective frequencybands, with widths in proportion to the bandwidths of the respectivefrequency bands.

Additional desirable properties for waveforms of high-bandwidth transmitelectromagnetic signals include a constant or near-constant modulus, toaccommodate physical constraints in peak power for radar transmitters;and good pulse compression properties, for target discrimination. Forexample, due to its performance under each of these criteria, the‘chirp’ or linear frequency modulated (LFM) electromagnetic signal iswidely used in known radar systems. However, because of itsnear-constant power spectral density over its bandwidth, the chirpelectromagnetic signal may cause interference with other systems thatare in the radar's operational range and operating at frequency bandswithin the radar's transmit signal waveform bandwidth. Thus, variousmethods have been developed for modifying the waveform of theelectromagnetic transmit signal to have minimal power spectral densityover these frequency bands.

SUMMARY OF THE INVENTION

One or more embodiments of the present invention transmit anelectromagnetic signal having a wide bandwidth constant-modulus waveformwith multiple frequency notches or stop-bands.

One or more embodiments of the present invention provide an iterativemethod for constructing an electromagnetic signal having aconstant-modulus, finite-duration waveform with a variable number ofnotches and stop-bands at user-specified frequencies within itsspectrum. A method of one or more embodiments of the present inventionmay be used to modify any initial electromagnetic signal having anyinitial waveform, such as a swept-frequency chirp-like waveform, to forma modified electromagnetic signal having a modified waverform and themodified waveform may have an arbitrary pulse envelope shape, includinga constant modulus envelope.

One or more embodiments of the present invention differ from priormethods for generating electromagnetic signals having a high-bandwidthwaveform with specified notches in that prior methods are non-iterative.The iterative nature of one or more embodiments of the present inventionmeans that frequency nulls may be achieved with arbitrary accuracy,meaning deeper and narrower frequency notches may be achieved than withprior methods. Unlike prior methods, one or more embodiments of thepresent invention are valid for wide stop-bands in short-durationpulses. One or more embodiments of the present invention may be used tocreate frequency nulls of arbitrary order at arbitrary frequencies.Thus, wide frequency stop-bands may be synthesized using higher-order orclosely-spaced frequency nulls, allowing flexibility for one or moreembodiments of the present invention in the Fourier magnitude of aconstructed electromagnetic transmit signal.

In at least one embodiment of the present invention a method is providedcomprising storing in a computer memory a first waveform for a zerothiteration; receiving in a computer memory a plurality of frequencyvalues; receiving in a computer memory a plurality of notch depths, suchthat each of the plurality of frequency values has a corresponding notchdepth of the plurality of notch depths; and receiving in a computermemory a pulse envelope function. The method may be further comprised ofrepeating the following steps for one or more iterations, starting withthe zeroth iteration, such that k=0 for the zeroth iteration: (a)modifying the first waveform for the kth iteration to create a secondwaveform for the kth iteration in computer memory, such that a Fouriertransform of the second waveform for the kth iteration is equal to zeroat each of the plurality of frequency values; (b) modifying the secondwaveform for the kth iteration to form a third waveform for the kthiteration in computer memory, such that the pulse envelope function ofthe third waveform for the kth iteration is equal to the pulse envelopefunction of the first waveform for the kth iteration; (c) if a stoppingcondition has not been met, storing the third waveform for the kthiteration as a first waveform for the (k+1) th iteration in computermemory, thereafter with k incremented by one, starting a kth iterationof the one or more iterations, and otherwise if the stopping conditionhas been met, stopping the one or more iterations, wherein the stoppingcondition is defined by a set of criteria stored in computer memory.

The first waveform for the zeroth iteration may be a swept-frequencychirp waveform of a specific bandwidth and duration. The pulse envelopefunction may be a constant modulus function.

In at least one embodiment, after the stopping condition has been met,the third waveform for the kth iteration has a plurality of stop bands,wherein each of the plurality of stop bands of the third waveform forthe kth iteration has a spectrum magnitude which is approximately zero.

In at least one embodiment, after the stopping condition has been met,the third waveform for the kth iteration includes a first notch at afirst frequency value, wherein the first notch has a Fourier transformvalue at the first frequency value which is approximately zero, whereinthe first notch is based on an order derivative of a Fourier transform,wherein the order derivative of the Fourier transform is greater than orequal to a first order derivative.

In at least one embodiment the first waveform of the zeroth iteration,called y₀, is a chirp waveform and is determined by a computer processorfrom the following equation:y ₀=exp(j(2πf ₀ t+βt ²)),0≦t≦Twherein f₀ is a carrier frequency of the first waveform), β is aparameter describing a bandwidth of the first waveform of the zerothiteration, and T is a duration of the first waveform of the zerothiteration y₀(n),

In at least one embodiment, the first waveform for the kth iteration,which is called y_(k)(t), is modified to form the second waveform forthe kth iteration, which is called z_(k)(t) by a computer processorusing the following equation:z _(k)(t)=y _(k)(t)Kexp(j2πf ₁ t),0≦t≦T;Wherein t is a continous time index; wherein

${K = {\frac{1}{T}{\int_{0}^{T}{{y_{k}(\tau)}{\exp\left( {{- {j2\pi}}\; f_{1}\tau} \right)}{\mathbb{d}\tau}}}}};$T is a duration of the first waveform for the kth iteration, y_(k)(t);and f is a specified null frequency, creating a single null at frequencyf₁.

In at least one embodiment, the first waveform for the kth iteration,which is called y_(k)(n), is modified to form the second waveform forthe kth iteration, which is called z_(k)(n) by a computer processorusing the following equation:z _(k)(n)=y _(k)(n)−Kexp(j2πf ₁ n),0≦n≦N−1wherein n is a discrete time index; wherein

${K = {\frac{1}{N}{\sum\limits_{n = 0}^{N - 1}{{y_{k}(n)}{\exp\left( {{- {j2\pi}}\; f_{1}n} \right)}}}}};$N is a number of points of the first waveform for the kth iteration,y_(k)(n); and f is a specified null frequency, creating a single null atfrequency f₁.

In at least one embodiment, the second waveform for the kth iteration,named z_(k)(t) is modified to form the third waveform for the kthiteration, named {tilde over (y)}_(k) _((t)) by the following equation:{tilde over (y)} _(k)(t)←exp(j<z _(k)(t)),0≦t≦T,wherein T is the duration of the first waveform for the zeroth iterationy₀(t), <(•) denotes the angle operation arctan (Im(•)/Re(•).

In at least one embodiment the stopping condition is that a total meansquared error between the third waveform for the kth iteration {tildeover (y)}_(k) _((t)) and the first waveform for the kth iterationy_(k)(t) which is computed as ∥{tilde over (y)}_(k)(t)−y_(k)(t)∥ by acomputer processor, is less than a threshold stored in computer memory.

In another embodiment, the first waveform of the zeroth iteration ischirp waveform specified as y₀(n) and is determined by a computerprocessor using the following equation:y ₀(n)=A(n)exp(j(2πf ₀ n+βn ²)),0≦n≦N−1  (1)wherein, A(n) is a pulse envelope function, f₀ is a carrier frequencyfor the first waveform for the kth iteration, β is a parameterdescribing the bandwidth of the waveform, and N is a number of samplesin the first waveform of the zeroth iteration.

In another embodiment, the first waveform for the kth iteration, whichis called y_(k) is modified to form the second waveform for the kthiteration, which is called z_(k), by using a computer processor todetermine the following equation:z _(k) ←y _(k) −QQ ^(H) y _(k)wherein Q is a matrix which is an orthogonalization of a matrix C whosecolumns consist of a plurality of vectors each of which is either adiscrete Fourier transform vector or a derivative vector of a orderderivative greater than or equal to a first order derivative, associatedwith the plurality of frequency values, given byC=[c ₁ ,c ₂ , . . . , c _(R)],wherein c₁, c₂, . . . , c_(R) specify a plurality of column vectors,such that the matrix C has r column vectors, and each column vector ofthe plurality of column vectors has the formc _(m)=[0,(−j2π)^(l−1) e ^(−j2πj) ^(m) , . . . , (−j2πi)^(l−1) e^(−j2πif) ^(m) , . . . , (−j2π(N−1))^(l−1) e ^(j2π(N−1)f) ^(m) ]wherein m is an index indicating a particular column vector of thematrix C; and wherein l is a an integer used to indicate a derivative ofan order greater than or equal to a first order derivative, such that lis greater or equal to one and Q^(H) is a matrix which is the Hermitianconjugate of Q.

In another embodiment, the second waveform for the kth iteration, namedz_(k)(n) is modified to form the third waveform for the kth iteration,named {tilde over (y)}_(k)(n) by using a computer processor to force thethird waveform for the kth iteration to have pulse envelope functionA(n) by the operation {tilde over (y)}_(k)(n)←A(n)exp(j<z_(k)(n)),0≦n≦N−1, where <(•) denotes the angle operation arctan (Im(•)/Re(•).

In another embodiment, the stopping condition is that a total meansquared error between the third waveform for the kth iteration {tildeover (y)}_(k)(n) and the first waveform for the kth iteration y_(k)(n),computed by a computer processor as ∥{tilde over (y)}_(k)(n)−y_(k)(n)∥₂,is less than a threshold stored in computer memory.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A shows a diagram of the spectrum magnitude of an electromagneticsignal having a standard chirp waveform with frequency swept over theinterval −0.1 Hz to 0.2 Hz;

FIG. 1B shows a diagram of the instantaneous frequency of anelectromagnetic signal having a standard chirp waveform in FIG. 1A, overa one thousand-sample duration;

FIG. 2 shows a diagram of a flow chart in accordance with a method of anembodiment of the present invention;

FIG. 3 shows an example of an embodiment of the present invention, formodifying an input electromagnetic signal having a chirp waveform toinclude a single frequency null at frequency f₁ and simultaneouslyexhibit a constant modulus;

FIG. 4 shows another example of an embodiment of the present invention,for modifying an input electromagnetic signal having a chirp waveform toinclude multiple frequency nulls and simultaneously exhibit a pulseenvelope shape of the initial input electromagnetic signal having thechirp waveform;

FIG. 5 shows a graph of error versus iteration number of an embodimentof the present invention;

FIG. 6A shows a graph of a spectrum magnitude of a modifiedelectromagnetic signal having a modified waveform, created by a methodof an embodiment of the present invention, by modifying theelectromagnetic signal having the waveform shown in FIG. 1A to include acreated null at frequency f₁=0 Hz;

FIG. 6B shows an instantaneous frequency of an electromagnetic signalhaving a waveform shown in FIG. 5A, over one thousand sample duration;

FIG. 7A shows an autocorrelation function of an electromagnetic signalhaving the waveform in FIG. 1A, for time lag from “−100” to “100”samples;

FIG. 7B shows an autocorrelation function of an electromagnetic signalhaving the waveform in FIG. 6A, for time lag from “−100” to “100”samples;

FIG. 8A shows a spectrum magnitude of an electromagnetic signal createdby an embodiment of the present invention, modifying the electromagneticsignal having the waveform shown in FIG. 1 a to include a created nullat frequency f₁=0.1 Hz;

FIG. 8B shows the instantaneous frequency of an electromagnetic signalhaving the waveform in FIG. 8A, over a one thousand-sample duration;

FIG. 9A shows a diagram of the spectrum magnitude of an electromagneticsignal having a waveform created by an embodiment of the presentinvention, modifying the electromagnetic signal having the waveform inFIG. 1A to include a 10^(th) order frequency null at frequency f₁=0.1025Hz;

FIG. 9B shows a diagram of the stop-band 901 a in FIG. 9A with a linearscale, for the frequencies 0.085 Hz to 0.12 Hz;

FIG. 9C shows a diagram of the stop-band 901 a in FIG. 9A with alogarithmic scale, for the frequencies 0.085 Hz to 0.12 Hz;

FIG. 10A shows a diagram of a spectrum magnitude of an electromagneticsignal having a waveform created by an embodiment of the presentinvention, modifying the electromagnetic signal having the waveform inFIG. 1A to include nine unevenly spaced nulls between frequency f₁=0.05Hz and f₉=0.055 Hz;

FIG. 10B shows a diagram of the stop-band 1001 a in FIG. 10A from thefrequency 0.04 Hz to frequency 0.065 Hz;

FIG. 11 shows a diagram of the spectrum magnitude of a standard chirpwaveform with frequency swept over the interval 0.1 Hz to 0.4 Hz;

FIG. 12A shows a diagram of the spectrum magnitude of a waveform createdby an embodiment of the present invention, modifying the waveform inFIG. 11 to include 18 nulls at the frequencies f_(m)=±{0.1500, 0.1506,0.01513, 0.1519, 0.1525, 0.1531, 0.1538, 0.1544, 0.1550};

FIG. 12B shows a diagram of the real value of the spectrum in FIG. 12Aat the stop-band 1201 b from the frequency 0.14 Hz to frequency 0.165Hz; and

FIG. 13 shows an apparatus in accordance with an embodiment of thepresent invention along with a target.

DETAILED DESCRIPTION OF THE DRAWINGS

FIG. 13 shows an apparatus 1300 in accordance with an embodiment of thepresent invention along with a target 1350. The apparatus 130 mayinclude a computer memory 1302, a computer display 1304, a computerprocessor 1306, a computer interactive device 1308, a transmitter 1310,and a receiver 1312. The computer memory 1302, the computer display1304, the computer interactive device 1308, the transmitter, and thereceiver 1312 may communicate with the computer processor 1306 such asthrough electrical communications lines, wireless communications, otherany other communications. The computer processor 1306 may be programmedby a computer program stored in computer memory 1302 to cause thetransmitter 1310 to transmit a signal towards the target 1350 throughthe airwaves 1314, and to receive a signal reflected from the target1350 via the airwaves 1316.

FIG. 1A shows a diagram 100. The diagram 100 includes an x axis labeled“Normalized Frequency (Hz)”, and a y axis labeled “(dB)”. The diagram100 includes a curve 101 which represents the spectrum magnitude|Y(f)|=|Σ_(n=0) ^(N−1)y(n)exp(−j2πfn)| of a standard chirpelectromagnetic signal waveform with frequency swept from −0.1 Hz to 0.2Hz. The chirp waveform may be represented by y(n)=A(n)exp(j(2πf₀n+βn²)),where A(n) is a pulse envelope function, and here f₀=−0.1 Hz, β=9.4210⁻⁴ _(x), and n=1→1000.

FIG. 1B shows a diagram 110. The diagram 110 includes an x axis labeled“Time (samples)”, and a y axis labeled “Normalized Frequency (Hz)”. Thediagram 100 includes a curve 111 which represents the instantaneousfrequency of the chirp waveform in FIG. 1A. The chirp waveform exhibitsa linearly sweeping frequency over the duration of the pulse.

FIG. 2 shows a flow chart 200, depicting a method of an embodiment ofthe present invention. At step 201, a first waveform for a zerothiteration (for k=0, may also be called a first iteration for a firstpass through a loop) is initialized by a computer processor, such ascomputer processor 1306 shown in FIG. 13. The first waveform may beinitialized by the computer processor 1306 by storing characteristics ofthe first wave form in a computer memory, such as computer memory 1302in FIG. 13, which may include a vector of data values representing adigitally sampled time series. This first waveform may be ahigh-bandwidth swept frequency signal, such as a chirp signal. At step202 an index k, may be initialized to 0 in computer memory 1302 for azeroth iteration (for k=0 or first iteration for first pass throughloop)_by the computer processor 1306.

At step 203, the first waveform for the kth iteration is modified into asecond waveform for the kth iteration by setting a plurality of valuesat a first plurality of frequency values to zero in computer memory 1302by the computer processor 1306. At step 204, the second waveform for thekth iteration is modified into a third waveform for the kth iteration byforcing the third waveform for the kth iteration to have constantmodulus, meaning the magnitude of the third waveform for the kthiteration which is a complex-valued signal must be constant over itsduration. The third waveform or characteristics of the third waveformfor the kth iteration are stored in computer memory 1302 by the computerprocessor 1306.

At step 205, the computer processor 1306 tests a stopping conditionstored in computer memory 1302. If the stopping condition is not met,step 206 is next executed by the computer processor 1306. At step 206 afirst waveform for the (k+1)th iteration is set equal to the thirdwaveform for the kth iteration in computer memory 1302 by the computerprocessor 1306. At step 207, k is incremented so, if k was “0” for thezeroth iteration, k is changed to “1” for the next iteration (or secondpass through the loop). The process then loops back to step 203.

If the stopping condition at step 205 produces a YES result, then theiterative process is stopped and the third waveform for the kthiteration is a constant modulus waveform with exact nulls and the thirdwaveform for the kth iteration is stored in computer memory 1302 by thecomputer processor 1306 at step 208. The computer processor 1306 causesan electromagnetic signal having the third waveform for the kthiteration to be transmitted out from the transmitter 1310 towards thetarget 1350 at step 209. At step 210, a return signal is received at thereceiver 1312, reflected back from the target 1350, as a result of thetransmitted signal along with noise and clutter. The return signal isprovided to the computer processor 1306 which processes the returnsignal and stores data based on the return signal in the computer memory1302 and displays data based on the return signal on computer display1304. The data based on the return signal, received at step 210, mayinclude characteristics of the return signal.

The following refers to a method of alternating projections inaccordance with a prior art method. We denote the notch-forming step asprojection operator P_(n) and the set of signals with notches at thefirst plurality of frequency values as S_(n). Similarly, enforcing theconstant-modulus property is denoted as projection P_(c), and the set ofconstant-modulus signals may be denoted as S_(c). However, becauseS_(c), is not a convex set, P_(c) does not form a convex projectionoperator, and thus the classical theory of projection-onto-convex sets(PoCS) [D. C. Youla and H. Webb. Image restoration by the method ofconvex projections: Part I—theory. IEEE Trans. Medical Imaging,MI-1(2):81-94, October 1982] may not be applied directly. Nevertheless,enforcing the constant-modulus property with P_(c) guaranteesdiminishing error between successive modified waveforms y_(k) [A. Leviand H. Stark, Image restoration by the method of generalized projectionswith applications to restorations from magnitude. In Proc. IEEE Int.Conf. Acoust., Speech, Signal Processing (ISCASSP), 1984].

Using these results, we can write the iterative method executed by thecomputer processor 1306 in FIG. 13 and as shown by flow chart 200 inFIG. 2 asy _(k+1) =P _(c) P _(n) y _(k) ,k 0, 1, 2,  (2)where y_(k) is the input to the k th iteration of the iterative methodshown by flow chart 200, and y₀ represents the first initializedwaveform. Define z_(k) asz _(k) =P _(n) y _(k)  (3)

Then z_(k) has an exact notch, but not the constant-modulus property. Byconstruction,y _(k) εS _(c) ,z _(k) εS _(n) for all k≧1.Then

$\begin{matrix}\begin{matrix}{{{{y_{k + 1} - z_{k + 1}}}} = {{{y_{k + 1} - {P_{n}y_{k}}}}}} \\{= {{{{P_{c}P_{n}y_{k}\mspace{14mu} P_{n}y_{k}}}}(5)}} \\{\leq {{{{y_{k} - {P_{n}y_{k}}}}}(6)}} \\{\leq {{{{y_{k} - z_{k}}}}(7)}}\end{matrix} & (4)\end{matrix}$Thus, the error does not increase with successive iterations of equation(1). Although this property does not guarantee the convergence ofequation (1), it does ensure that the waveforms produced do not diverge.In practice, one or more embodiments of the present invention based onthe above iterative method of alternating projections convergesreliably.

FIG. 3 shows a flow chart 300, depicting a method of another embodimentof the present invention, creating a null at a specified frequency f₁ inan arbitrary initial waveform. The method shown by flow chart 300 may beimplemented by the computer processor 1306 as programmed by a computerprogram stored in the computer memory 1302. At step 301, a firstwaveform for a zeroth iteration (for k=0, may be called a firstiteration instead of a zeroth iteration for first pass through a loop),y₀(n) is initialized by the computer processor 1306. The first waveformfor the zeroth iteration (k=0) of step 301 may be a first chirp waveformhaving equation of:exp(j(2πf ₀ t+βt ²)),0≦t≦T; or exp(j(2πf ₀ n+βn ²)),0≦n≦N−1,  (8)wherein t is a continous time index, wherein n is a discrete time index,and the equation in (7) with t is used for a continous time index caseand the equation with n is used for a discrete time index case, whereinf₀ is a carrier frequency of the first chirp waveform, β is a parameterdescribing a bandwidth of the first chirp waveform, and T is a durationof the first chirp waveform y₀(n), (i.e. an example of the firstwaveform for the zeroth iteration of steps 201 or 301). The computerprocessor 1306 may store in computer memory 1302 the results of, thevalues of, the variables of, and any characteristics of equation (7) toinitialize the first chirp waveform (an example of first waveform forthe zeroth iteration). The variable k may be set to zero at step 301. Atstep 302, the first chirp waveform (an example of a first waveform forthe zeroth iteration is modified by the computer processor 1306 to forma second waveform for the zeroth iteration z₀(n), through the operationz _(k)(t)=y _(k)(t)Kexp(j2πf ₁ t),0≦t≦T; or z _(k)(n)=y _(k)(n)Kexp(j2πf₁ n),0≦n≦N−1,  (9)wherein t is a continous time index, wherein n is a discrete time index,and the equation in (8) with t is used for a continous time index caseand the equation with n is used for a discrete time index case.Here

${K = {\frac{1}{T}{\int_{0}^{T}{{y_{k}(\tau)}{\exp\left( {{- {j2\pi}}\; f_{1}\tau} \right)}{\mathbb{d}\tau}}}}};$T is the duration of the first chirp waveform and f is a specified nullfrequency, creating a single null at frequency f₁. The computerprocessor 1306 may store in computer memory 1302 the results of, thevalues of, the variables of, and any characteristics of equation (8). Ingeneral, the computer processor 1306 may store in computer memory 1302the results of, the values of, the variables of, and any characteristicsof any of the equations provided in the present application.

At step 303, the computer processor 1306 forces the second waveform forzeroth iteration (k=0, or can be instead called first iteration forfirst pass through loop), z₀(n) to be modified to form a third waveformfor the zeroth iteration, {tilde over (y)}_(k) having a constant modulusby the operation {tilde over (y)}_(k)(t)←exp(j<z_(k)(t)), 0≦t≦T, whereinT is the duration of the first chirp waveform y₀(n), <(•) denotes theangle operation arctan (Im(•)/Re(•). The computer processor 1306 maystore in computer memory 1302 the results of, the values of, thevariables of, and any characteristics of the equation executed at step303.

At step 304, the total mean squared error between the third waveform forthe zeroth iteration {tilde over (y)}_(k) and the first waveform for thezeroth iteration y_(k) is computed as ∥{tilde over (y)}_(k)−y_(k)∥ bythe computer processor 1306, and is compared with a threshold Δ storedin computer memory 1302. If the stopping condition is not met, thecomputer processor 1306 is programmed to execute step 305. At step 305,the first waveform for the next iteration (i.e. the k+1 th iteration),is set equation to the third waveform for the current iteration (i.e.the kth iteration). Thereafter K is incremented, i.e. set to k+1 at step306 and the flow returns to step 302 for another iteration.

If the stopping condition at step 304 produces a YES result, then theiterative process is stopped and the third waveform for the kthiteration, {tilde over (y)}_(k) is a constant modulus waveform withexact nulls and the third waveform for the kth iteration {tilde over(y)}_(k) is stored in computer memory 1302 by the computer processor1306 and the computer processor 1306 causes an electromagnetic signalhaving the third waveform for the kth iteration, {tilde over (y)}_(k) tobe transmitted out from the transmitter 1310 towards the target 1350 atstep 307. At step 308, a return signal is received at the receiver 1312,reflected back from the target 1350, as a result of the transmittedsignal along with noise and clutter. The return signal is provided tothe computer processor 1306 which processes the return signal and storesdata based on the return signal in the computer memory 1302 and displaysdata based on the return signal on computer display 1304. The data basedon the return signal, received at step 308, may include characteristicsof the return signal.

The third waveform for the kth iteration, {tilde over (y)}_(k), of theelectromagnetic signal transmitted out transmitter 1310 at step 307, hasan approximately constant-modulus chirp-like waveform with a null atfrequency f₁. The computer processor 1306 may store characteristics ofthe third waveform for the kth iteration {tilde over (y)}_(k) and maysupply those characteristics to the transmitter 1310 and may cause thetransmitter 1310 to output an electromagnetic signal having the storedcharacteristics of the third waveform for the kth iteration {tilde over(y)}_(k) to the airwaves 1314 towards the target 1350. Theelectromagnetic signal may be reflected and/or modified by the target1350 and received back via the airwaves 1316 at an input of the receiver1312.

Instead of a chirp signal for the first waveform of the zerothiteration, in at least one embodiment of the present invention, thefirst waveform for the zeroth iteration may be a waveform with anyarbitrary envelope shape, such as A(n) as will be described withreference to a FIG. 4 embodiment. FIG. 4 shows a flow chart 400,depicting a method in accordance with another embodiment of the presentinvention. The embodiment shown by FIG. 4, creates a number of frequencynulls at specified frequencies f_(m), m=1→M, where M is the number ofnulls to be created, in an electromagnetic signal having an initialstandard chirp waveform, while preserving the pulse envelope A(n) of theinitial waveform. At step 401, a first chirp waveform, y₀(n) (or firstwaveform for a kth iteration, with k=0) is initialized by the followingequation:y ₀(n)=A(n)exp(j(2πf ₀ n+βn ²)),0≦n≦N−1  (10)wherein, where A(n) is a pulse envelope function, f₀ is a carrierfrequency for the first waveform for the kth iteration, β is a parameterdescribing the bandwidth of the waveform, and N is the number of samplesin the waveform. The first chirp waveform or first waveform for the kthiteration, with k=0) may be initialized by the computer processor 1306determining and storing characteristics, values, and/or variables of theequation (9) in the computer memory 1302. At step 402, the firstwaveform for the kth iteration, with k=0, is modified to form a secondwaveform for the kth iteration, with k=0, which has M nulls, each of adiffering order L_(m), m=1→M, at the frequencies f_(m), m=1→M andcharacteristics, values, and/or variables of this second waveform forthe kth iteration with k=0 may be determined by and stored in computermemory 1302 by the computer processor 1306. Creating these nulls isequivalent to satisfying the system of equations

$\begin{matrix}{{{{\frac{\partial^{l}}{\partial f}{Y\left( f_{m} \right)}} - \left( {\frac{\partial^{l}}{\partial f}\left\lbrack {\sum\limits_{n = 0}^{N - 1}{{y(n)}{\exp\left( {{j2\pi}\; f\;\overset{\_}{n}} \right)}}} \right\rbrack} \right)_{f = f_{m}}},0}{{0 = {\leq l \leq L_{m}}},{1 \leq m \leq M},}} & (11)\end{matrix}$where

$\frac{\partial^{l}\left( . \right)}{\partial f}$denotes the lth derivative with respect to f, orC ^(H) y=0  (12)where C is an N×R matrix with R=Σ_(m=0) ^(M)L_(m), of the formC=[c ₁ ,c ₂ , . . . c _(R)],  (13)where the column vectors are of the formc _(m)=[0,(−j2π)^(l−1) e ^(−j2πf) ^(m) , . . . , (−j2πi)^(l−1) e^(−j2πif) ^(m) , . . . , (−j2π(N−1))^(l−1) e ^(−j2π(N−1)f) ^(m) ],  (14)If all the specified nulls are first-order, this reduces toc _(m)=[1,e ^(−j2πf) ^(m) ,e ^(−j4πf) ^(m) , . . . , e ^(−j2πif) ^(m) ,. . . , e ^(−j2(N−1)πf) ^(m) ]^(T).  (15)

Note (•)^(H) is the Hermitian conjugate operation, and 0 is theR-element zero vector. Given a waveform y, the waveform x closest to asolution y of equation (11) is given by x=Py, where P is the orthogonalprojection operator given byP=Iε(C ^(H) C)⁻¹ C ^(H)  (16)where I is the N×N identity matrix. To avoid numerical ill-conditioningresulting from the inverse operation in equation (15), theorthogonalization Q of C may be constructed by the computer processor1306 using an orthogonalization step, which may be programmed incomputer memory 1302 and implemented by the computer processor 1306 suchas Gram-Schmidt orthogonalization [D. Bau III and L. N. Trefethen,Numerical Linear Algebra, pp. 50-59. Siam 1997], so that the columns ofQ spans the same space as the columns of C, and Q^(H)Q=I. Then P mayalternatively be constructed asP=IQQ ^(H).  (17)Thus the constraint Q^(H)y=0 is equivalent to the constraint in equation(11), and may be implemented asz _(k) ←y _(k) −QQ ^(H) y _(k)  (18)at step 402. The variables, values, matrices and/or characteristics ofthe equations (13) and (14) may be determined by the computer processor1306 and stored in computer memory 1302.

At step 403 the second waveform for the kth iteration, z_(k), ismodified to form a third waveform for the kth iteration {tilde over(y)}_(k), by the computer processor 1306 by forcing the third waveformfor the kth iteration {tilde over (y)}_(k) to have pulse envelopefunction A(n) by the operation {tilde over(y)}_(k)(n)←A(n)exp(j<z_(k)(n)), 0≦N−1, where <(•) denotes the angleoperation arctan (Im( )/Re( ), and the variables, values andcharacteristics of the result are stored in computer memory 1302. Atstep 404, the computer processor 1306 determines the total mean squarederror between the third waveform for the kth iteration {tilde over(y)}_(k) and the first waveform for the kth iteration y_(k), computed as∥{tilde over (y)}_(k)−y_(k)∥₂, and compares the total mean squared errorwith a threshold Δ stored in computer memory 1302. If the stoppingcondition is not met, at step 405, the first waveform for the (k+1)thiteration is set equal to the third waveform for the kth iteration, andthen thereafter k is incremented at step 406 to k=k+1; and then flowreturns to step 402.

If the stopping condition at step 404 produces a YES result, then theiterative process is stopped and the third waveform for the kthiteration, {tilde over (y)}_(k) is a constant modulus waveform withexact nulls and the third waveform for the kth iteration {tilde over(y)}_(k) is stored in computer memory 1302 by the computer processor1306 at step 407. The third waveform for the kth iteration, {tilde over(y)}_(k), is a chirp-like waveform with the pulse envelope A(n) of theinitial waveform, and with nulls at frequencies f_(m), m=1→M. Thecomputer processor 1306 causes an electromagnetic signal having thethird waveform for the kth iteration, {tilde over (y)}_(k) to betransmitted out from the transmitter 1310 towards the target 1350 atstep 407. At step 408, a return signal is received at the receiver 1312,reflected back from the target 1350, as a result of the transmittedsignal along with noise and clutter. The return signal is provided tothe computer processor 1306 which processes the return signal and storesdata based on the return signal in the computer memory 1302 and displaysdata based on the return signal on computer display 1304. The data basedon the return signal, received at step 408, may include characteristicsof the return signal.

FIG. 5 shows a diagram 500. The diagram 500 includes an x axis labeled“Iteration”, and a y axis labeled “Error (dB) (decibels)”. The diagram500 includes a curve 501, representing the maximum value of the vectorC^(H){tilde over (y)} as in equation (11), where {tilde over (y)} is thethird waveform for the kth iteration, when the stopping condition hasbeen met, for the first fifty iterations of the method shown in FIG. 4of one example trial of an embodiment of the present invention. Thepresent invention in one or more embodiments demonstrates fastexponential-like convergence.

FIG. 6A shows a diagram 600. The diagram 600 includes an x axis labeled“Normalized Frequency (Hz)” (Hz—hertz), and a y axis labeled “(dB)”. Thediagram 600 includes a curve 601 representing the spectrum magnitude|Y(f)|=|Σ_(n=0) ^(N−1)y(n)exp(−j2πfn)| of an example outputelectromagnetic signal y(n) having the third waveform of the kthiteration, when the stopping condition has been met for FIG. 3embodiment example, output from the transmitter 1310 to the airwaves1314 towards the target 1350 in an embodiment of the present invention.The third waveform of the kth iteration, when the stopping condition ofFIG. 3 met, y(n) was created with the method of the embodimentrepresented by flow chart 300 in FIG. 3, with y₀(n) equal to the chirpfunction in FIG. 1A and FIG. 1B, and f₁=0 Hz. The curve 601 includesregion 601 a, showing a deep notch at f₁=0 Hz.

FIG. 6B shows a diagram 610. The diagram 610 includes an x axis labeled“Time (samples)” and a y axis labeled “Normalized Frequency (Hz)”. Thediagram 610 includes a curve 611, representing the instantaneousfrequency of an electromagnetic signal having the notched waveform inFIG. 6A, created by the computer processor 1306 in accordance with anembodiment of the present invention. The instantaneous frequency of thecreated electromagnetic signal having the notched waveform isapproximately linear, resembling the instantaneous frequency of theinitial waveform depicted in FIG. 1B, but with a small added oscillatorycomponent.

FIG. 7A shows a diagram 700. The diagram 700 includes an x axis labeled“Time (samples)” and a y axis labeled “Autocorrelation Value”. Thediagram 700 includes a curve 701, representing the autocorrelationfunction R_(yy)(s)=Σ_(n=0) ^(N−1)y(n)y(n−s) for real-valued signals, forthe standard chirp waveform depicted in FIG. 1A and FIG. 1B. The widthof the main peak of the autocorrelation function, and the height of theside peaks, are important parameters for evaluating the resolution ofradar systems utilizing chirp and chirp-like waveforms. FIG. 7B shows adiagram 710. The diagram 710 includes an x axis labeled “Time (samples)”and a y axis labeled “Autocorrelation Value”. The diagram 710 includes acurve 711, representing the autocorrelation function R_(yy)(s)=Σ_(n=0)^(N−1)y(n)y(n−s) for the notched chirp waveform created by the presentinvention, depicted in FIG. 6A and FIG. 6B. The curve 711 is similar tocurve 701 in FIG. 7A, suggesting that the notched chirp waveform createdby an embodiment of the present invention has similar resolutionproperties to the standard chirp waveform without the notched frequencyproperty added by the present invention. FIG. 8A shows a diagram 800.The diagram 800 includes an x axis labeled “Normalized Frequency (Hz)”and a y axis labeled “(dB)”. The diagram 800 includes a curve 801representing the spectrum magnitude |Y(f)|=|Σ_(n=0)^(N−1)y(n)exp(−j2πfn)| of an example output electromagnetic signal y(n)from the transmitter 1310 to the airwaves 1314 of an embodiment of thepresent invention. The waveform y(n) is the third waveform for the kthiteration when the stopping condition is satisfied with the method ofthe embodiment represented by the flow chart 300 in FIG. 3, with y₀(n)equal to the chirp function in FIG. 1A and FIG. 1B, and f₁=0.1 Hz. Thecurve 801 includes region 801 a, showing a deep notch at f₁=0.1 Hz.

FIG. 8B shows a diagram 810. The diagram 810 includes an x axis labeled“Time (samples)” and a y axis labeled “Normalized Frequency (Hz)”. Thediagram 810 includes a curve 811, representing the instantaneousfrequency of the notched waveform in FIG. 8A, created by an embodimentof the present invention. The instantaneous frequency of the createdwaveform is approximately linear, resembling the instantaneous frequencyof the initial waveform depicted in FIG. 1B, but with a small addedoscillatory component. FIG. 9A shows a diagram 900. The diagram 900includes an x axis labeled “Normalized Frequency (Hz)” and a y axislabeled “(dB)”. The diagram 900 includes a curve 901 representing thespectrum magnitude |Y(f)|=|Σ_(n=0) ^(N−1)y(n)exp(−j2πfn)| of an exampleoutput electromagnetic signal y(n) from the transmitter 1310 to theairwaves 1314 of an embodiment of the present invention. The waveformy(n) is the third waveform for the kth iteration when the stoppingcondition of FIG. 4 is satisfied and was created with the embodimentrepresented by 400 in FIG. 4, with y₀(n) equal to the chirp function inFIG. 1A and FIG. 1B, and a 10^(th) order null at f_(m)=0.1025 Hz,1≦m≦10. The curve 901 includes region 901 a, showing a deep stop-bandcentered at frequency 0.1025 Hz.

FIG. 9B shows a diagram 910. The diagram 910 includes an x axis labeled“Normalized Frequency (Hz)” and a y axis labeled “Magnitude”. Thediagram 910 includes curve 901 in FIG. 9A, plotted with alinearly-scaled y axis, from frequency 0.085 Hz to frequency 0.12 Hz.The curve 901 includes region 901 a, showing the stop-band created by anembodiment of the present invention.

FIG. 9C shows a diagram 920. The diagram 920 includes an x axis labeled“Normalized Frequency (Hz)” and a y axis labeled “(dB)”. The diagram 920includes curve 901 in FIG. 9A, plotted with a logarithmically-scaled yaxis, from frequency 0.085 Hz to frequency 0.12 Hz. The curve 901includes region 901 a, showing the stop-band created by an embodiment ofthe present invention.

FIG. 10A shows a diagram 1000. The diagram 1000 includes an x axislabeled “Normalized Frequency (Hz)” and a y axis labeled “(dB)”. Thediagram 1000 includes a curve 1001 representing the spectrum magnitude|Y(f)|=|Σ_(n=0) ^(N−1)y(n)exp(−j2πfn)| of an example outputelectromagnetic signal y(n) from the transmitter 1310 to the airwaves1314 of an embodiment of the present invention. The waveform of FIG. 10Ay(n) is the third waveform for the kth iteration when the stoppingcondition of FIG. 4 is satisfied and was created with the embodimentrepresented by 400 in FIG. 4, with y₀(n) equal to the chirp function inFIG. 1A and FIG. 1B, and f_(m)={0.0500, 0.0504, 0.0510, 0.517, 0.0525,0.0532, 0.0539, 0.0546, 0.055} Hz. The curve 1001 includes region 1001a, showing a deep stop-band from frequency 0.05 Hz to frequency 0.055Hz. The stop-band 1001 a exhibits approximate equiripple behavior due tothe chosen uneven spacing of the frequency nulls.

FIG. 10B shows a diagram 1010. The diagram 1010 includes an axis labeled“Normalized Frequency (Hz)” and a y axis labeled “(dB)”. The diagram1010 includes curve 1001 in FIG. 9A, from frequency −0.04 Hz tofrequency 0.065 Hz. The curve 1001 includes region 1001 a, showing thestop-band created by the present invention.

FIG. 11 shows a diagram 1100. The diagram 1100 includes an x axislabeled “Normalized Frequency (Hz)” and a y axis labeled “(dB)”. Thediagram 1100 includes a curve 1101, representing the spectrum magnitude|Y(f)|=|Σ_(n=0) ^(N−1)y(n)exp(−j2πfn)| of a standard chirp waveformy(n), with frequency swept from 0.1 Hz to 0.4 Hz.

FIG. 12A shows a diagram 1200. The diagram 1200 includes an x axislabeled “Normalized Frequency (Hz)” and a y axis labeled “(dB)”. Thediagram 1200 includes a curve 1201, representing the spectrum magnitude|Y(f)|=|Σ_(n=0) ^(N−1)y(n)exp(−j2πfn)| of an example output y(n) of thepresent invention. The waveform y(n) shown in FIG. 12A, is the thirdwaveform for the kth iteration when the stopping condition is satisfiedin FIG. 4, and was created with the embodiment represented by 400 inFIG. 4, with y₀(n) equal to the standard chirp waveform depicted in FIG.10, and notched frequencies f_(m)={0.1500, 0.1506, 0.01513, 0.1519,0.1525, 0.1531, 0.1538, 0.1544, 0.1550}. The curve 1101 includes region1101 a, showing a notched band in the negative frequency range createdby the present invention, and region 1101 b, showing a notched band inthe positive frequency range created by the present invention.

FIG. 12B shows a diagram 1210. The diagram 1210 includes an x axislabeled “Normalized Frequency (Hz)” and a y axis labeled “(dB)”. Thediagram 1210 includes curve 1211, representing the real value of thewaveform spectrum over the frequency range 0.14 Hz to 0.165 Hz. Thecurve 1211 includes region 1211 a, showing the real value of thestop-band whose magnitude is shown by 1201 b in FIG. 11A. Region 1211 ashows the deep notches created by the present invention are maintainedwhen the real part of the waveform is taken—this is due to the chosenfrequency notches having symmetry about the frequency 0 Hz. Thisdemonstrates that real-valued waveforms with deep notches and deepstop-bands may be created using an embodiment of the present invention.

Although the invention has been described by reference to particularillustrative embodiments thereof, many changes and modifications of theinvention may become apparent to those skilled in the art withoutdeparting from the spirit and scope of the invention. It is thereforeintended to include within this patent all such changes andmodifications as may reasonably and properly be included within thescope of the present invention's contribution to the art.

We claim:
 1. A method for creating a radar transmit waveform comprising:storing in a computer memory a first waveform for a zeroth iteration;receiving in a computer memory a plurality of frequency values;receiving in a computer memory a plurality of notch depths, such thateach of the plurality of frequency values has a corresponding notchdepth of the plurality of notch depths; receiving in a computer memory apulse envelope function; repeating the following steps for one or moreiterations, starting with the zeroth iteration, such that k=0 for thezeroth iteration: modifying the first waveform for the kth iteration tocreate a second waveform for the kth iteration in computer memory, suchthat a Fourier transform of the second waveform for the kth iteration isequal to zero at each of the plurality of frequency values; modifyingthe second waveform for the kth iteration to form a third waveform forthe kth iteration in computer memory, such that the pulse envelopefunction of the third waveform for the kth iteration is equal to thepulse envelope function of the first waveform for the kth iteration; ifa stopping condition has not been met, storing the third waveform forthe kth iteration as a first waveform for the (k+1) th iteration incomputer memory, thereafter with k incremented by one, starting a kthiteration of the one or more iterations, and otherwise if the stoppingcondition has been met, stopping the one or more iterations, wherein thestopping condition is defined by a set of criteria stored in computermemory.
 2. The method of claim 1 further comprising: if the stoppingcondition has been met, transmitting an electromagnetic signal havingthe third waveform for the kth iteration out from a transmitter towardsa target, and receiving a return electromagnetic signal from the target.3. The method of claim 1 wherein the first waveform for the zerothiteration is a swept-frequency chirp waveform of a specific bandwidthand duration.
 4. The method of claim 1 wherein the pulse envelopefunction is a constant modulus function.
 5. The method of claim 1wherein after the stopping condition has been met, the third waveformfor the kth iteration has a plurality of stop bands; wherein each of theplurality of stop bands of the third waveform for the kth iteration hasa spectrum magnitude which is approximately zero.
 6. The method of claim1 wherein after the stopping condition has been met, the third waveformfor the kth iteration includes a first notch at a first frequency value;wherein the first notch has a Fourier transform value at the firstfrequency value which is approximately zero and wherein the first notchis based on an order derivative of a Fourier transform, wherein theorder derivative of the Fourier transform is greater than or equal to afirst order derivative.
 7. The method of claim 1 wherein the firstwaveform of the zeroth iteration, called y₀, is a chirp waveform and isdetermined by a computer processor from the following equation:y ₀=exp(j(2πf ₀ t+βt ²)),0≦t≦T wherein f₀ is a carrier frequency of thefirst waveform, β is a parameter describing a bandwidth of the firstwaveform of the zeroth iteration, and T is a duration of the firstwaveform of the zeroth iteration y₀(n).
 8. The method of claim 1 whereinthe first waveform for the kth iteration, which is called y_(k)(t), ismodified to form the second waveform for the kth iteration, which iscalled z_(k)(t) by a computer processor using the following equation:z _(k)(t)=y _(k)(t)Kexp(j2πf₁ t),0≦t≦T; Wherein t is a continous timeindex; wherein${K = {\frac{1}{T}{\int_{0}^{T}{{y_{k}(\tau)}{\exp\left( {{- {j2\pi}}\; f_{1}\tau} \right)}{\mathbb{d}\tau}}}}};$T is a duration of the first waveform for the kth iteration, y_(k)(t);and f₁ is a specified null frequency, creating a single null atfrequency f₁.
 9. The method of claim 1 wherein the first waveform forthe kth iteration, which is called y_(k)(n), is modified to form thesecond waveform for the kth iteration, which is called z_(k)(n) by acomputer processor using the following equation:z _(k)(n)=y _(k)(n)−Kexp(j2πf ₁ n),0≦n≦N−1 Wherein n is a discrete timeindex; wherein${K = {\frac{1}{N}{\sum\limits_{n = 0}^{N - 1}{{y_{k}(n)}{\exp\left( {{- {j2\pi}}\; f_{1}n} \right)}}}}};$N is a number of points of the first waveform for the kth iteration,y_(k)(n); and f₁ is a specified null frequency, creating a single nullat frequency f₁.
 10. The method of claim 1 wherein the second waveformfor the kth iteration, named z_(k)(t) is modified to form the thirdwaveform for the kth iteration, named {tilde over (y)}_(k)(t) by thefollowing equation:{tilde over (y)} _(k)(t)←exp(j<z _(k)(t)),0≦t≦T, wherein T is theduration of the first waveform for the zeroth iteration y₀(t), <(•)denotes the angle operation arctan(Im(•)/Re(•).
 11. The method of claim1 wherein the stopping condition is that a total mean squared errorbetween the third waveform for the kth iteration {tilde over (y)}_(k)(t)and the first waveform for the kth iteration y_(k)(t) which is computedas ∥{tilde over (y)}_(k)(t)−y_(k)(t)∥ by a computer processor, is lessthan a threshold stored in computer memory.
 12. The method of claim 1wherein the first waveform of the zeroth iteration is chirp waveformspecified as y₀(n) and is determined by a computer processor using thefollowing equation:y ₀(n)=A(n)exp(j(2πf ₀ n+β ²)),0≦n≦N−1  (19) wherein, A(n) is a pulseenvelope function, f₀ is a carrier frequency for the first waveform forthe kth iteration, β is a parameter describing the bandwidth of thewaveform, and N is a number of samples in the first waveform of thezeroth iteration.
 13. The method of claim 1 wherein the first waveformfor the kth iteration, which is called y_(k) is modified to form thesecond waveform for the kth iteration, which is called zk, by using acomputer processor to determine the following equation:z _(k) ←y _(k) −QQ ^(H) y _(k) wherein Q is a matrix which is anorthogonalization of a matrix C whose columns consist of a plurality ofvectors each of which is either a discrete Fourier transform vector or aderivative vector of a order derivative greater than or equal to a firstorder derivative, associated with the plurality of frequency values,given byC=[c ₁ , c ₂ , . . . , c _(R)], wherein c₁, c₂, . . . , c_(R) specify aplurality of column vectors, such that the matrix C has r columnvectors, and each column vector of the plurality of column vectors hasthe formc _(m)=[0,(−j2π)^(l−1) e ^(−j2πf) ^(m) , . . . , (−j2πi)^(l−1) e^(−j2πif) ^(m) , . . . , (−j2π(N−1))^(l−1) e ^(−j2π(N−1)f) ^(m) ]wherein m is an index indicating a particular column vector of thematrix C; and wherein l is a an integer used to indicate a derivative ofan order greater than or equal to a first order derivative, such that lis greater or equal to one and Q^(H) is a matrix which is the Hermitianconjugate of Q.
 14. The method of claim 1 wherein the second waveformfor the kth iteration, named z_(k)(n) is modified to form the thirdwaveform for the kth iteration, named {tilde over (y)}_(k)(n), by usinga computer processor to force the third waveform for the kth iterationto have pulse envelope function A(n) by the operation {tilde over(y)}_(k)(n)←A(n)exp(j<z_(k)(n)), 0≦n≦N−1, where <(•) denotes the angleoperation arctan (Im(•)/Re(•)).
 15. The method of claim 1 wherein thestopping condition is that a total mean squared error between the thirdwaveform for the kth iteration {tilde over (y)}_(k)(n), and the firstwaveform for the kth iteration y_(k)(n), computed by a computerprocessor as ∥{tilde over (y)}_(k)(n)−y_(k)(n)∥₂, is less than athreshold stored in computer memory.
 16. An apparatus for creating aradar transmit waveform comprising: a computer processor; and a computermemory; wherein the computer processor is configured to communicate withthe computer memory; wherein the computer processor is programmed by acomputer program to: store in the computer memory a first waveform for azeroth iteration; store in the computer memory a plurality of frequencyvalues; store in the computer memory a plurality of notch depths, suchthat each of the plurality of frequency values has a corresponding notchdepth of the plurality of notch depths; and store in the computer memorya pulse envelope function; and wherein the computer processor isprogrammed by a computer program to repeat the following steps for oneor more iterations, starting with the zeroth iteration, such that k=0for the zeroth iteration: (a) modify the first waveform for the kthiteration to create a second waveform for the kth iteration in thecomputer memory, such that a Fourier transform of the second waveformfor the kth iteration is equal to zero at each of the plurality offrequency values; (b) modify the second waveform for the kth iterationto form a third waveform for the kth iteration in the computer memory,such that the pulse envelope function of the third waveform for the kthiteration is equal to the pulse envelope function of the first waveformfor the kth iteration; and (c) if a stopping condition has not been met,store the third waveform for the kth iteration as a first waveform forthe (k+1) th iteration in the computer memory, thereafter with kincremented by one, starting a kth iteration of the one or moreiterations, and otherwise if the stopping condition has been met,stopping the one or more iterations, wherein the stopping condition isdefined by a set of criteria stored in the computer memory.
 17. Theapparatus of claim 16 wherein the first waveform for the zerothiteration is a swept-frequency chirp waveform of a specific bandwidthand duration.
 18. The apparatus of claim 16 wherein the pulse envelopefunction is a constant modulus function.
 19. The apparatus of claim 16wherein after the stopping condition has been met, the third waveformfor the kth iteration has a plurality of stop bands; wherein each of theplurality of stop bands of the third waveform for the kth iteration hasa spectrum magnitude which is approximately zero.
 20. The apparatus ofclaim 16 wherein after the stopping condition has been met, the thirdwaveform for the kth iteration includes a first notch at a firstfrequency value; wherein the first notch has a Fourier transform valueat the first frequency value which is approximately zero and wherein thefirst notch is based on an order derivative of a Fourier transform,wherein the order derivative of the Fourier transform is greater than orequal to a first order derivative.
 21. The apparatus of claim 16 whereinthe first waveform of the zeroth iteration, called y₀, is a chirpwaveform and is determined by the computer processor from the followingequation:y ₀=exp(j(2πf ₀ t+βt ²)),0≦t≦T wherein f₀ is a carrier frequency of thefirst waveform, β is a parameter describing a bandwidth of the firstwaveform of the zeroth iteration, and T is a duration of the firstwaveform of the zeroth iteration y₀(n).
 22. The apparatus of claim 16wherein the first waveform for the kth iteration, which is calledy_(k)(t), is modified to form the second waveform for the kth iteration,which is called z_(k)(t) by the computer processor using the followingequation:z _(k)(t)=y _(k)(t)Kexp(j2πf ₁ t),0≦t≦T; Wherein t is a continous timeindex; wherein${K = {\frac{1}{T}{\int_{0}^{T}{{y_{k}(\tau)}{\exp\left( {{- {j2\pi}}\; f_{1}\tau} \right)}{\mathbb{d}\tau}}}}};$T is a duration of the first waveform for the kth iteration, y_(k)(t);and f₁ is a specified null frequency, creating a single null atfrequency f₁.
 23. The apparatus of claim 16 wherein the first waveformfor the kth iteration, which is called y_(k)(n), is modified to form thesecond waveform for the kth iteration, which is called z_(k)(n) by thecomputer processor using the following equation:z _(k)(n)=y _(k)(n)−Kexp(j2πf ₁ n),0≦n≦N−1 Wherein n is a discrete timeindex; wherein${K = {\frac{1}{N}{\sum\limits_{n = 0}^{N - 1}{{y_{k}(n)}{\exp\left( {{- {j2\pi}}\; f_{1}n} \right)}}}}};$N is a number of points of the first waveform for the kth iteration,y_(k)(n); and f₁ is a specified null frequency, creating a single nullat frequency f₁.
 24. The apparatus of claim 16 wherein the secondwaveform for the kth iteration, named z_(k)(t) is modified to form thethird waveform for the kth iteration, named {tilde over (y)}_(k)(t) bythe following equation:{tilde over (y)} _(k)(t)←exp(j<z _(k)(t)),0≦t≦T, wherein T is theduration of the first waveform for the zeroth iteration y₀(t), <(•)denotes the angle operation arctan (Im(•)/Re(•).
 25. The apparatus ofclaim 16 wherein the stopping condition is that a total mean squarederror between the third waveform for the kth iteration {tilde over(y)}_(k)(t) and the first waveform for the kth iteration y_(k)(t) whichis computed as ∥{tilde over (y)}_(k)(t)−y_(k)(t)| by the computerprocessor, is less than a threshold stored in the computer memory. 26.The apparatus of claim 16 wherein the first waveform of the zerothiteration is chirp waveform specified as y₀(n) and is determined by acomputer processor using the following equation:y ₀(n)=A(n)exp(j(2πf ₀ n+βn ²)),0≦n≦N−1  (20) wherein, A(n) is a pulseenvelope function, f₀ is a carrier frequency for the first waveform forthe kth iteration, β is a parameter describing the bandwidth of thewaveform, and N is a number of samples in the first waveform of thezeroth iteration.
 27. The apparatus of claim 16 wherein the firstwaveform for the kth iteration, which is called y_(k) is modified toform the second waveform for the kth iteration, which is called z_(k),by using the computer processor to determine the following equation:z _(k) ←y _(k) −QQ ^(H) y _(k) wherein Q is a matrix which is anorthogonalization of a matrix C whose columns consist of a plurality ofvectors each of which is either a discrete Fourier transform vector or aderivative vector of a order derivative greater than or equal to a firstorder derivative, associated with the plurality of frequency values,given byC=[c ₁ , c ₂ , . . . , c _(R)], wherein c₁, c₂, . . . , c_(R) specify aplurality of column vectors, such that the matrix C has r columnvectors, and each column vector of the plurality of column vectors hasthe formc _(m)=[0,(−j2π)^(l−1) e ^(−j2πf) ^(m) , . . . , (−j2πi ^(l−1) e^(−j2πif) ^(m) , . . . , (−j2π(N−1))^(l−1) e ^(−j2π(N−1)f) ^(m) ]wherein m is an index indicating a particular column vector of thematrix C; and wherein l is a an integer used to indicate a derivative ofan order greater than or equal to a first order derivative, such that lis greater or equal to one and Q^(H) is a matrix which is the Hermitianconjugate of Q.
 28. The apparatus of claim 16 wherein the secondwaveform for the kth iteration, named z_(k)(n) is modified to form thethird waveform for the kth iteration, named {tilde over (y)}_(k)(n), byusing the computer processor to force the third waveform for the kthiteration to have pulse envelope function A(n) by the operation {tildeover (y)}_(k)(n)←A(n)exp(j<z_(k)(n)), 0≦n≦N−1, where <(•) denotes theangle operation arctan(Im(•)/Re(•)).
 29. The apparatus of claim 16wherein the stopping condition is that a total mean squared errorbetween the third waveform for the kth iteration {tilde over(y)}_(k)(n), and the first waveform for the kth iteration y_(k)(n),computed by a computer processor as ∥{tilde over (y)}_(k)(n)−y_(k)(n)∥₂,is less than a threshold stored in computer memory.
 30. The apparatus ofclaim 16 further comprising a transmitter, and a receiver which areconfigured to communicate with the computer processor; and wherein thecomputer processor is programmed by a computer program to cause anelectromagnetic signal having the third waveform for the kth iterationto be transmitted out from the transmitter towards a target, and thecomputer processor is configured to process a return electromagneticsignal from the target received at the receiver.